When I changed the pump only on ours the clamp that held the fuel line was not adequate to seal the fuel line to the injectors. The symptom of the issue was that with key on the fuel pressure (gage on fuel rail) would show up to 50 psi and the pump would shut off and the pressure bled off to zero. Pump would not stay running but would turn on again when the ignition key was cycled. Once i secured the pressure side of the pump output properly everything was fine. The other symptom was a long cranking period before the engine would start (which made me realize that something was still wrong). I am still not sure what signal is required to keep the pump running. It may be oil pressure or something else. Storey Wilson did a video that also showed that a relay could be intermittent but that seems less likely. Perhaps Ray has an answer to this question which may help you solve this.